Affine variety sketching

Question

Can somebody help me to understand how to sketch an affine variety in $R^3$, when $V((x-2)(x^2-y),y(x^2-y),(z+1)(x^2-y))$?

Do I need to equate each term to zero and solve the system?

Answer 1

We can break it down by noting that $V(fg)=V(f)\cup V(g)$ and $V(f,g)=V(f)\cap V(g)$.

Working out these intersections and unions in your case gives $$V(x^2-y)\cup (V(x-2)\cap V(y)\cap V(z+1))$$ so we'll be left with a parabola running across the $z$-axis along with a single point $(2,0,-1)$.